a r e m a r k a b l e p e r s o n o n m a n y accounts. author of the book u n d e r review. Roy Most obviously, he was a first-rate mathdied shortly after earning a B.A. in anematician a n d bore m u c h responsibility thropology. Alan earned a B.S. in physics, for bringing m o d e r n algebra into the worked as an engineer, and has since current mathematics curriculum. He passed away. Nancy e a r n e d a J.D. a n d was especially famous for his work o n works :is a lawyer. nonassociative algebras, o n division alI am reviewing the J u n e 2006 revised edition of this biography. With m o d e r n gebras, and on R i e m a n n matrices. Perp u b l i s h i n g methods it is easy to correct haps less k n o w n , he bore major responsibility for persuading the U.S. typographical errors and to take into ac g o v e r n m e n t to support basic research c o u n t n e w l y available source materials. in mathematics. Even before the naNancy Albert has d o n e just that. In adtional i m p o r t a n c e of science a n d mathdition, it is remarkable just h o w Nancy ematics was u n d e r l i n e d by Russia's Albert was able to transpose her legal l a u n c h of the first satellite, "sputnik," in abilities and produce such a precise and 1957, Adrian Albert played a key role accurate description of Adrian Albert's in establishing a n d increasing the mathematics, of the honors he received, a m o u n t of ONR a n d NSF research supa n d of his interactions with the federal port for active mathenmticians at all levg o v e r n m e n t and the other m e m b e r s of els of seniority. Some of this i n f l u e n c e the scientific establishment of his day. certainly must have b e e n b a s e d o n his Her description of the history of the d e f e n s e work during a n d after World University of Chicago, especially its DeWar II, in particular his effective use of partment of Mathematics, is accurate algebraic m e t h o d s in cwptology. a n d to the point. She is precise and inPerhaps k n o w n mostly to those w h o cisive w h e n she discusses the social seth a d direct personal contact with him, ting of both her father and her mother, Adrian Albert put a lot of time. thought, his c o n n e c t i o n s with the University of and effort into e n c o u r a g i n g a n d Chicago a n d the other m e m b e r s of the s m o o t h i n g the way for students a n d D e p a r t m e n t of Mathematics there, and y o u n g researchers. O n m a n y occasions, (rather delicately) the way he m a n a g e d s o m e of which are m e n t i o n e d in this despite anti-Semitic attitudes that lasted biography, he arranged financial supinto the late 1950s. She was well acport for students to e n a b l e full-time q u a i n t e d with the many mathematicians study. He e n c o u r a g e d w o m e n in the of her father's age and a bit younger, study of mathematics, a n d to the extent a n d that certainly contributed to the then possible, he facilitated their career quality of the book. Also, recently she paths. The biography contains several h a d access to some relevant AMS instances of this, and there were in fact archives. quite a few others. Adrian Albert usuThis biography has already b e e n really did this sort of thing in a very w a r m v i e w e d by Lance Small in the AMS Xohearted way: he w o u l d arrange the b e n tices ( D e c e m b e r 2005) and by Philip efit a n d then gleefully surprise the Davis in the SIAM News (June 2006). recipient with the good news. Lance Small is a n algebraist w h o started (Abraham) Adrian Albert, often nickhis graduate studies at the University of n a m e d '% Cubed," was b o r n in 1905 Chicago toward the e n d of Adrian Ala n d grew u p in Chicago, received his bert's career. I defer to his description B.S. in mathematics from the University of Adrian's mathematical results beof Chicago in 1926, a n d e a r n e d his cause I work in areas ( g e o m e t W a n d Ph.D. from Leonard Dickson at the Unianalysis) more or less orthogonal to versity of Chicago in 1928. After two Adrian's research. There is o n e n o n years as Instructor at Columbia Univer- trivial point of historical disagreement sity he r e t u r n e d to the University of b e t w e e n Lance Small's review and Chicago as Assistant Professor in 1931, Nancy Albert's book. It c o n c e r n s Adrian Full Professor in 1941, Math I ) e p a r t m e n t Albert's efforts to generate an offer from Chair 1958-1962, a n d Dean of Physical the University of Chicago to Nathan Sciences 1962-1971. He passed away in J a c o b s o n a n d the possibility that these 1972. efforts were initially i m p e d e d by antiAdrian Albert married Frieda Davis Semitic attitudes. Nancy Albert has clarin 1927. They had two sons, Alan a n d ified this situation, with better docuRoy, a n d a daughter, Nancy, w h o is the m e n t a t i o n a n d a better a r r a n g e m e n t of

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the text, in the J u n e 2006 revised edition of her book. Adrian Albert a n d my father were close friends since their u n d e r g r a d u a t e clays in the 1920s. W h e n Adrian Albert died he left a list of y o u n g mathematicians to be invited to each take some b o o k s flom his libraw, a n d I was o n the list. S o m e h o w that last gesture was typical of his generosity toward his students a n d y o u n g e r colleagues. As is clear at this point, this review is written from the v i e w p o i n t of a friend, student, a n d colleague, rather than from the v i e w p o i n t of a historian of mathematics or a mathematical critic. With that caveat, I definitely r e c o m m e n d this biography to all mathematicians interested in the interplay b e t w e e n mathematics a n d public policy, and especially those in pure or a p p l i e d algebra a n d those w h o had contact with the Dep a r t m e n t of Mathematics at the University of Chicago a n y time from the 1920s through the 1960s. Department of Mathematics University of California Berkeley, CA, 94720-3840 USA e-mail: jawolf@math, berkeley.edu

Saunders Mac Lane. A Mathematical Autobiography by Saunders M a c Lane, WELLESLEY, MASSACHUSETTS, A. K. PETERS, 2004, 358 P., $39.00, HARDCOVER, ISBN 1-56881-150-0 REVIEWED BY HENRY E. HEATHERLY

~aunders Mac Lane (1909-2005) ~was o n e of the e p o c h - m a k i n g ~: J * mathematicians of the 20th century. He knew- a n d interacted with m a n y of the o u t s t a n d i n g figures of 20th t e n tury mathematics. Add to this the wellk n o w n lucidity of his expositions, his p r o f o u n d insight into the nature of mathematics, a n d his experience with scientific organizations a n d mathematical centers of excellence, a n d one comes to the p a g e s of his autobiograp h y with high expectations. These ex-

pectations a r e - - f o r the most p a r t - - w e l l met. I never met Mac Lane, but having read all or parts of several of his books a n d many of his expository papers and letters to the :Votices, I feel I am acquainted with Mac Lane the mathematician a n d Mac Lane the c o m m e n t a tor on the state of the mathematics profession. The b o o k consists of fifteen parts (64 chapters), ranging over Mac Lane's long a n d active life. It b e g i n s with family background, early years, a n d his formal education. Alter u n d e r g r a d u a t e years at Yale, Mac Lane b e g a n graduate studies in the mathenmtics d e p a m n e n t at the University of Chicago, w h e r e he had a fellowship. Overall he f o u n d his year there to be disappointing, especially because he saw n o possibility of doing a dissertation o n logic at Chicago. He did write a master's thesis, l e a r n e d how to play bridge, and met Dorothy Jones (a graduate student in e c o n o m i c s w h o m he later married) in that year. He w o n an Institute for International Education fellowship a n d d e c i d e d to c o n t i n u e his studies in G6ttingen, w h e r e he could write a thesis o n logic, w h i c h had become his m a i n area of interest. G6ttiny e n in 1931 was the p r e m i e r mathematical center in the world. There Mac Lane b e g a n a thesis u n d e r the direction of Paul Bemays, Hilbert's chief assistant, a n d b e c a m e active in the mathematical life of late W e i m a r G e r m a n y . Regulations issued by the new' Nazi regime in 1933 forced Bernays to leave GOttingen a n d Mac Lane officially finished his doctoral thesis u n d e r Herm a n n Weyl. In July 1933 Mac Lane took his doctoral exams a n d two days later got m a r r i e d - - t h e n back to the U.S.A. with a postdoctoral Sterling Felk)wship at Yale. Professor Oystein Ore supervised all postdoctoral fellows in mathematics at Yale then, a n d Ore worked in algebra. So Mac Lane did research in valuation theory u n d e r Ore's direction, "with considerable enthusiasm," as Mac Lane recalls. Over the next several years Mac Lane held faculty positions at Harvard (where he was a B e n j a m i n Peirce instructor), Comell, the University of Chicago, a n d then back to Harvard as an assistant professor, in 1938. There he collaborated with Garrett Birkhoff to produce their influential book, A Survey o f Moderrt Algebra ( 1941 ). It was the first American u n d e r g r a d u a t e textbook

that presented the then n e w abstract ideas of Emmy N o e t h e r a n d B. L. van der \Vacrden. At Harvard, Mac Lane b e g a n to direct Ph.D. dissertati(ms. His frst student w a s Irving Kaplansky, w h o completed a dissertation. "Maximal Fields With Valuations," in 1941. Over the next sixty years Mac Lane directed Ph.D. dissertation research for at least forty students, most of them at the University of Chicago. The dissertations were in m a n y different areas: algebra, logic, algebraic topology, category theory, topos theory, a n d theoretical c o m p u t e r science. He also co-directed two Ph.D. dissertations on the histo W of mathematics. Among his m a n y students was a future Fields Medal winner, J o h n Thompson. A n o t h e r of his students, David Eisenbud, provided the preface for this book. A list of forty of his doctoral students is given in the Mathematics Genealogy Project website, with Steven Awodey listed as Mac Lane's last student, in 1997. A p h o t o g r a p h of Awodey and Mac Lane is given o n page 314 of the autobiography. It is clear that Mac Lane greatly e n j o y e d his experience in the g u i d a n c e of graduate students, calling it "a s p l e n d i d activity." While o n a o n e - s e m e s t e r research leave in 19ql Mac Lane started a collaboration with Samuel Eilenberg. This b e g a n with work o n group extensions, homology, and c o h o m o l o g y that eventually led to their seminal work o n category theory. T h e latter is probably the mathematical w o r k for which Mac Lane is best known. Mac Lane's collaboration with Eilenberg resulted in twenty-four joint papers in the period 1941-1955. These papers r a n g e d over several major mathematical ideas: h o m o t o p y structure of spaces; homological algebra (e.g., the c o h o m o l o g y of groups); category theory; a n d simplicial sets. The basic notions of category, functor, a n d natural transformation a p p e a r e d in definitive form in their 1945 paper. In the spring of 1943 Mac Lane became involved in mathematical work related to the war effort. He joined the Applied Mathematics G r o u p at Columbia. There he w o r k e d o n p r o b l e m s that came from the Air Force, e.g., g u n n e r y guidance and pursuit cmwes. The Columbia group had m a n y outstanding mathematicians besides Mac Lane, including Kaplansky, Eilenberg, Hassler

Whitney, Marston Morse, a n d Magnus Hestenes. At night Mac Lane and Eilenberg w o u l d work o n their o w n mathematical proMems in Eilenberg's apartmerit. Category theory took shape during this period. A G u g g e n h e i m Fellowship allowed Mac Lane to s p e n d the academic year 1947-48 in Europe. In Paris he a t t e n d e d Leray's lectures o n algebraic topology a n d got to k n o w Armand Borel a n d Serre. Next Mac Lane went to Z0rich, w h e r e he stayed for a longer p e r i o d of time to w o r k with Heinz H o p f at the Swiss Federal Technical Institute. Mac Lane says it had b e e n Hopf's ideas that originally started him and Eilenberg w o r k i n g o n the c o h o m o l o g y of groups. After Z0rich, Mac Lane visited several other mathematical centers in G e r m a n y , the Netherlands, Belgium, a n d Great Britain. In Oxford he b e g a n collaboration with ,1. H. C. Whitehead o n algebraic topology, the resuhs of which were p u b l i s h e d in 1950. Mac Lane a n d W h i t e h e a d had tentative plans to do further joint research, but to Mac Lane's regret this "never came to fruition." Returning from Europe he did not go back to Harvard, but instead a c c e p t e d a professorship at the University of Chicago. This was the Chicago of what has b e e n called "The Stone Age," because Marshall Stone had recently bec o m e c h a i r m a n there and was b u i l d i n g the mathematics d e p a r t m e n t to what arguably b e c a m e the premier o n e in the U.S.A. a n d p e r h a p s the world. The senior faculty were Albert, Chern, Stone, Well, Z y g m u n d , and Mac Lane. J u n i o r m e m b e r s i n c l u d e d Kaplansky, Halmos. Segal, and Spanier. In 1952 Stone s t e p p e d d o w n as chairman and Mac Lane took his place, serving from 1952 to 1958. He states that "the c h a i r m a n ' s job wits a t r o u b l e d one, with its major policy issues a n d assorted bureaucratic regulations." He was frustrated by administrative troubles ranging from keeping an efficient d e p a m n e n t secreta W to losing Andrd Weil from the facuhy because the administration gave Weft n o t h i n g to c o u n t e r an offer fiom the Institute for A d v a n c e d Study. During the period 1952-1958, no n e w t e n u r e d app o i n t m e n t s were made to the department, although some excellent n o n t e n u r e d a n d tenure track a p p o i n t m e n t s were made. Mac Lane c o n s i d e r e d his term as c h a i r m a n "a p r o l o n g a t i o n of the

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Stone Age that c o n t i n u e d , but did not expand, that tradition." In 1949 Mac Lane was elected to the National A c a d e m y of Sciences (NAS), leading to several decades of service with that organization. In 1958 he was elected a m e m b e r of the Council of the NAS a n d in 1959 he was n a m e d chairm a n of the editorial b o a r d for the Proceedings of the Academy, a position he held for eight years. He served two terms, 1973-1981, as vice president of the NAS. His m a i n activity during that term in office was in m a n a g i n g the Report Review Committee. This committee reviewed reports o n matters of highlevel science policy for various g o v e r n m e n t agencies. He was elected president of the Mathematical Association of America (MAA) for 1952-1953, a n d of the American Mathematical Society (AMS) in 1972 for a two-year term. He also was a n editor of the Bulletin a n d the Transactions of the AMS. In 1973-1983 he served as a m e m b e r of the National Science Board. Mac Lane had a long a n d distinguished career in the m a k i n g of science policy at the highest level. This activity a n d his mathematical research c o n t i n u e d until late in his long life. Mac Lane collaborated o n research with m a n y mathematicians over the s p a n of his career. Besides the ones already m e n t i o n e d , some of his coauthors were Otto Schilling, Alfred Clifford, a n d V. W. Adkisson. He coauthored s e v e n papers with Schilling in the period 1939-1943, o n algebras a n d n u m b e r fields. With Adkisson he w o r k e d o n geometric topology, a n d with Clifford, o n g r o u p theory. From early in his career Mac Lane's interest in mathematical logic led him to concerns a b o u t the p h i l o s o p h y of mathematics. He f o u n d "most of the subject misdirected a n d felt that o n e should be able to describe more accurately what really is there in mathematics." In 1983 he gave a series of lectures at the University of Minnesota, which he later o r g a n i z e d into a book, Mathematics, Form, a n d Function (1986). He also wrote several papers o n the general subject of the p h i l o s o p h y of mathematics a n d what mathematics i s - - o r should b e - - a b o u t . Throughout his autobiography Mac Lane w e a v e s in p e r s o n a l items a n d details a b o u t his p e r s o n a l life. He gives

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lively descriptions of his wide-ranging travels, of people met, a n d of sights seen. There are m a n y photos of mathematicians a n d of Mac Lane's family. Mac Lane gives us several examples of his well k n o w n h u m o r o u s verses. Here is a s a m p l e from o n e he read at the b a n q u e t for the International Conference o n Category Theory in Coimhra, Portugal, in 1999. Sam Eilenberg said just o n e paper will do To i n t r o d u c e categorical notions so new We'll write it so well these ideas for to sell A n d p u b l i s h it promptly the story to tell. He e n d s the book with a p o i g n a n t thought: "All told, mathematics was a great career choice for me." There are subjects on which I wish Mac Lane h a d said more, e.g., his personal relationships with Eilenberg and Stone a n d the period while he was c h a i r m a n at Chicago. While Mac Lane makes brief m e n t i o n of his brother Gerald, he does not m e n t i o n that Gerald was also a mathematician, an analyst w h o was a professor at Rice University a n d Purdue. There is no m e n t i o n of any mathematical interplay b e t w e e n the two brothers. The g e n e r a l reader will find a few chapters of the autobiography to be tough g o i n g mathematically. This is especially true for chapters twelve through fourteen, where some mathematical sophistication is n e e d e d to digest the c o n c e p t s introduced and disc u s s e d - - e . g . , crossed product algebras, covariant functors, Horn, a n d Ext. This caveat aside, a general scientific audience s h o u l d find Mac Lane's b o o k both informative a n d delightful. O n a technical note, for those wishing to use this a u t o b i o g r a p h y as a source of information. the format of the index will be disappointing. O n l y the names of people are listed in the index. So, looking up items such as category theory, the National Science Foundation, or Harvard, is m a d e m o r e difficult. For these reasons, a n d to gain an external viewpoint, o n e might h o p e that a talented and mathematically knowledgeable biograp h e r will n o w come along a n d give us a n o t h e r b o o k - l e n g t h view of this extraordinary man, Saunders Mac Lane. As a s u p p l e m e n t to the autobiogra-

p h y u n d e r review, tile references b e l o w s h o u l d prove of interest to the reader. Note that the two AMS v o l u m e s c o n t a i n five articles by Mac Lane, each well worth reading. REFERENCES

P. Duren (ed.), A Century of Mathematics in America, Parts fl and III, Providence, R.l., Amer. Math. Soc., 1989. D. Eisenbud, Encountering Saunders Mac Lane, Focus 25 (2005), 5-7. I. Kaplansky (ed.), Saunders Mac Lane Selected Papers, New York, Springer-Verlag, 1979. J. MacDonald, Saunders Mac Lane, 19092005, Focus 25 (2005), 4. S. Mac Lane, Mathematics at Gottingen Under the Nazis, Notices Amer. Math. Soc. 42(1995), 1134-1138. Mathematics Department University of Louisiana at Lafayette Lafayette, LA 70504-1010 U.S.A. e-mail: [email protected] mcheat [email protected]

Alfred Tarsld: Life and Logic by Anita B u r d m a n Feferman a n d Solomon Feferman CAMBRIDGE UNIVERSITY PRESS (OCTOBER 2004) ISBN: 0521802407, HARDCOVER, Vl + 425 PP, $34.99 REVIEWED BY KRZYSZTOF R. APT

f y o u ask a well-educated p e r s o n for the n a m e s of the three most promii~l n e n t logicians in the twentieth century, he will u n d o u b t e d l y come u p with G6del, Tarski, a n d Turing. While wellk n o w n biographies of the first a n d last have b e e n p u b l i s h e d - - a n d their tragic deaths attracted peoples' sympathetic a t t e n t i o n - - n o a c c o u n t of Tarski's life has a p p e a r e d until recently. The b o o k u n d e r review fills this gap excellently by providing a marvelously readable, informative, a n d gossipy account of his life a n d work. The b o o k is far from being a dry a c c o u n t of Tarski's achievements: o n the contrary. Tarski was a b o n - v i v a n t par excellence, and by delv-